LMFDB - Embedded newform 1840.3.c.b.1151.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(1151,1840)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1840.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.14
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.43

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.31883i q^{3} -2.23607 q^{5} -7.99843i q^{7} -2.01461 q^{9} +O(q^{10})$	$q-3.31883i q^{3} -2.23607 q^{5} -7.99843i q^{7} -2.01461 q^{9} +0.813639i q^{11} -5.17383 q^{13} +7.42112i q^{15} +14.1254 q^{17} -27.2204i q^{19} -26.5454 q^{21} +4.79583i q^{23} +5.00000 q^{25} -23.1833i q^{27} -1.17458 q^{29} -21.5683i q^{31} +2.70033 q^{33} +17.8850i q^{35} -26.7216 q^{37} +17.1710i q^{39} -29.7840 q^{41} -77.4205i q^{43} +4.50481 q^{45} +26.2177i q^{47} -14.9749 q^{49} -46.8798i q^{51} +20.2733 q^{53} -1.81935i q^{55} -90.3399 q^{57} +18.6197i q^{59} -63.5020 q^{61} +16.1137i q^{63} +11.5690 q^{65} -6.92224i q^{67} +15.9165 q^{69} -48.2310i q^{71} +41.4860 q^{73} -16.5941i q^{75} +6.50783 q^{77} +47.0335i q^{79} -95.0728 q^{81} +151.514i q^{83} -31.5854 q^{85} +3.89823i q^{87} -54.9939 q^{89} +41.3825i q^{91} -71.5815 q^{93} +60.8667i q^{95} -25.0525 q^{97} -1.63917i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 120 q^{9}+O(q^{10}) $ 56 * q - 120 * q^9	$ 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) $ 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	− 3.31883i	− 1.10628i	−0.833090	−	0.553138i	$-0.813430\pi$
$3$	− 3.31883i	− 1.10628i	0.833090	−	0.553138i	$-0.186570\pi$
$4$	0	0
$5$	−2.23607	−0.447214
$5$	−2.23607	−0.447214
$6$	0	0
$7$	− 7.99843i	− 1.14263i	−0.820730	−	0.571316i	$-0.806433\pi$
$7$	− 7.99843i	− 1.14263i	0.820730	−	0.571316i	$-0.193567\pi$
$8$	0	0
$9$	−2.01461	−0.223846
$10$	0	0
$11$	0.813639i	0.0739672i	0.999316	+	0.0369836i	$0.0117749\pi$
$11$	0.813639i	0.0739672i	−0.999316	+	0.0369836i	$0.988225\pi$
$12$	0	0
$13$	−5.17383	−0.397987	−0.198993	−	0.980001i	$-0.563767\pi$
$13$	−5.17383	−0.397987	−0.198993	+	0.980001i	$0.563767\pi$
$14$	0	0
$15$	7.42112i	0.494741i
$16$	0	0
$17$	14.1254	0.830906	0.415453	−	0.909615i	$-0.363623\pi$
$17$	14.1254	0.830906	0.415453	+	0.909615i	$0.363623\pi$
$18$	0	0
$19$	− 27.2204i	− 1.43265i	−0.697765	−	0.716327i	$-0.745822\pi$
$19$	− 27.2204i	− 1.43265i	0.697765	−	0.716327i	$-0.254178\pi$
$20$	0	0
$21$	−26.5454	−1.26407
$22$	0	0
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	− 23.1833i	− 0.858641i
$28$	0	0
$29$	−1.17458	−0.0405028	−0.0202514	−	0.999795i	$-0.506447\pi$
$29$	−1.17458	−0.0405028	−0.0202514	+	0.999795i	$0.506447\pi$
$30$	0	0
$31$	− 21.5683i	− 0.695752i	−0.937541	−	0.347876i	$-0.886903\pi$
$31$	− 21.5683i	− 0.695752i	0.937541	−	0.347876i	$-0.113097\pi$
$32$	0	0
$33$	2.70033	0.0818281
$34$	0	0
$35$	17.8850i	0.511001i
$36$	0	0
$37$	−26.7216	−0.722205	−0.361102	−	0.932526i	$-0.617599\pi$
$37$	−26.7216	−0.722205	−0.361102	+	0.932526i	$0.617599\pi$
$38$	0	0
$39$	17.1710i	0.440283i
$40$	0	0
$41$	−29.7840	−0.726438	−0.363219	−	0.931704i	$-0.618322\pi$
$41$	−29.7840	−0.726438	−0.363219	+	0.931704i	$0.618322\pi$
$42$	0	0
$43$	− 77.4205i	− 1.80048i	−0.435396	−	0.900239i	$-0.643392\pi$
$43$	− 77.4205i	− 1.80048i	0.435396	−	0.900239i	$-0.356608\pi$
$44$	0	0
$45$	4.50481	0.100107
$46$	0	0
$47$	26.2177i	0.557823i	0.960317	+	0.278912i	$0.0899737\pi$
$47$	26.2177i	0.557823i	−0.960317	+	0.278912i	$0.910026\pi$
$48$	0	0
$49$	−14.9749	−0.305609
$50$	0	0
$51$	− 46.8798i	− 0.919211i
$52$	0	0
$53$	20.2733	0.382514	0.191257	−	0.981540i	$-0.438744\pi$
$53$	20.2733	0.382514	0.191257	+	0.981540i	$0.438744\pi$
$54$	0	0
$55$	− 1.81935i	− 0.0330791i
$56$	0	0
$57$	−90.3399	−1.58491
$58$	0	0
$59$	18.6197i	0.315588i	0.987472	+	0.157794i	$0.0504382\pi$
$59$	18.6197i	0.315588i	−0.987472	+	0.157794i	$0.949562\pi$
$60$	0	0
$61$	−63.5020	−1.04102	−0.520509	−	0.853856i	$-0.674258\pi$
$61$	−63.5020	−1.04102	−0.520509	+	0.853856i	$0.674258\pi$
$62$	0	0
$63$	16.1137i	0.255773i
$64$	0	0
$65$	11.5690	0.177985
$66$	0	0
$67$	− 6.92224i	− 0.103317i	−0.998665	−	0.0516585i	$-0.983549\pi$
$67$	− 6.92224i	− 0.103317i	0.998665	−	0.0516585i	$-0.0164507\pi$
$68$	0	0
$69$	15.9165	0.230674
$70$	0	0
$71$	− 48.2310i	− 0.679310i	−0.940550	−	0.339655i	$-0.889690\pi$
$71$	− 48.2310i	− 0.679310i	0.940550	−	0.339655i	$-0.110310\pi$
$72$	0	0
$73$	41.4860	0.568301	0.284150	−	0.958780i	$-0.408288\pi$
$73$	41.4860	0.568301	0.284150	+	0.958780i	$0.408288\pi$
$74$	0	0
$75$	− 16.5941i	− 0.221255i
$76$	0	0
$77$	6.50783	0.0845173
$78$	0	0
$79$	47.0335i	0.595361i	0.954666	+	0.297680i	$0.0962129\pi$
$79$	47.0335i	0.595361i	−0.954666	+	0.297680i	$0.903787\pi$
$80$	0	0
$81$	−95.0728	−1.17374
$82$	0	0
$83$	151.514i	1.82547i	0.408549	+	0.912736i	$0.366035\pi$
$83$	151.514i	1.82547i	−0.408549	+	0.912736i	$0.633965\pi$
$84$	0	0
$85$	−31.5854	−0.371593
$86$	0	0
$87$	3.89823i	0.0448072i
$88$	0	0
$89$	−54.9939	−0.617909	−0.308954	−	0.951077i	$-0.599979\pi$
$89$	−54.9939	−0.617909	−0.308954	+	0.951077i	$0.599979\pi$
$90$	0	0
$91$	41.3825i	0.454753i
$92$	0	0
$93$	−71.5815	−0.769694
$94$	0	0
$95$	60.8667i	0.640702i
$96$	0	0
$97$	−25.0525	−0.258273	−0.129137	−	0.991627i	$-0.541221\pi$
$97$	−25.0525	−0.258273	−0.129137	+	0.991627i	$0.541221\pi$
$98$	0	0
$99$	− 1.63917i	− 0.0165572i
$100$	0	0
$101$	30.9030	0.305971	0.152985	−	0.988228i	$-0.451111\pi$
$101$	30.9030	0.305971	0.152985	+	0.988228i	$0.451111\pi$
$102$	0	0
$103$	− 34.8866i	− 0.338705i	−0.985556	−	0.169353i	$-0.945832\pi$
$103$	− 34.8866i	− 0.338705i	0.985556	−	0.169353i	$-0.0541677\pi$
$104$	0	0
$105$	59.3573	0.565308
$106$	0	0
$107$	106.550i	0.995797i	0.867235	+	0.497898i	$0.165895\pi$
$107$	106.550i	0.995797i	−0.867235	+	0.497898i	$0.834105\pi$
$108$	0	0
$109$	−67.3792	−0.618158	−0.309079	−	0.951036i	$-0.600021\pi$
$109$	−67.3792	−0.618158	−0.309079	+	0.951036i	$0.600021\pi$
$110$	0	0
$111$	88.6843i	0.798957i
$112$	0	0
$113$	28.1039	0.248707	0.124354	−	0.992238i	$-0.460314\pi$
$113$	28.1039	0.248707	0.124354	+	0.992238i	$0.460314\pi$
$114$	0	0
$115$	− 10.7238i	− 0.0932505i
$116$	0	0
$117$	10.4233	0.0890876
$118$	0	0
$119$	− 112.981i	− 0.949420i
$120$	0	0
$121$	120.338	0.994529
$122$	0	0
$123$	98.8478i	0.803641i
$124$	0	0
$125$	−11.1803	−0.0894427
$126$	0	0
$127$	220.126i	1.73328i	0.498938	+	0.866638i	$0.333724\pi$
$127$	220.126i	1.73328i	−0.498938	+	0.866638i	$0.666276\pi$
$128$	0	0
$129$	−256.945	−1.99182
$130$	0	0
$131$	− 110.353i	− 0.842387i	−0.906971	−	0.421194i	$-0.861611\pi$
$131$	− 110.353i	− 0.842387i	0.906971	−	0.421194i	$-0.138389\pi$
$132$	0	0
$133$	−217.721	−1.63700
$134$	0	0
$135$	51.8394i	0.383996i
$136$	0	0
$137$	−152.144	−1.11054	−0.555270	−	0.831670i	$-0.687385\pi$
$137$	−152.144	−1.11054	−0.555270	+	0.831670i	$0.687385\pi$
$138$	0	0
$139$	− 71.3279i	− 0.513150i	−0.966524	−	0.256575i	$-0.917406\pi$
$139$	− 71.3279i	− 0.513150i	0.966524	−	0.256575i	$-0.0825941\pi$
$140$	0	0
$141$	87.0120	0.617106
$142$	0	0
$143$	− 4.20963i	− 0.0294380i
$144$	0	0
$145$	2.62644	0.0181134
$146$	0	0
$147$	49.6990i	0.338088i
$148$	0	0
$149$	−41.2821	−0.277061	−0.138531	−	0.990358i	$-0.544238\pi$
$149$	−41.2821	−0.277061	−0.138531	+	0.990358i	$0.544238\pi$
$150$	0	0
$151$	− 201.085i	− 1.33169i	−0.746091	−	0.665844i	$-0.768072\pi$
$151$	− 201.085i	− 1.33169i	0.746091	−	0.665844i	$-0.231928\pi$
$152$	0	0
$153$	−28.4572	−0.185995
$154$	0	0
$155$	48.2282i	0.311150i
$156$	0	0
$157$	189.355	1.20608	0.603040	−	0.797711i	$-0.293956\pi$
$157$	189.355	1.20608	0.603040	+	0.797711i	$0.293956\pi$
$158$	0	0
$159$	− 67.2834i	− 0.423166i
$160$	0	0
$161$	38.3591	0.238255
$162$	0	0
$163$	73.1734i	0.448917i	0.974484	+	0.224458i	$0.0720613\pi$
$163$	73.1734i	0.448917i	−0.974484	+	0.224458i	$0.927939\pi$
$164$	0	0
$165$	−6.03812	−0.0365946
$166$	0	0
$167$	268.575i	1.60823i	0.594472	+	0.804117i	$0.297361\pi$
$167$	268.575i	1.60823i	−0.594472	+	0.804117i	$0.702639\pi$
$168$	0	0
$169$	−142.231	−0.841606
$170$	0	0
$171$	54.8386i	0.320693i
$172$	0	0
$173$	123.885	0.716096	0.358048	−	0.933703i	$-0.383442\pi$
$173$	123.885	0.716096	0.358048	+	0.933703i	$0.383442\pi$
$174$	0	0
$175$	− 39.9921i	− 0.228527i
$176$	0	0
$177$	61.7955	0.349127
$178$	0	0
$179$	− 153.717i	− 0.858753i	−0.903126	−	0.429376i	$-0.858733\pi$
$179$	− 153.717i	− 0.858753i	0.903126	−	0.429376i	$-0.141267\pi$
$180$	0	0
$181$	−155.542	−0.859347	−0.429674	−	0.902984i	$-0.641371\pi$
$181$	−155.542	−0.859347	−0.429674	+	0.902984i	$0.641371\pi$
$182$	0	0
$183$	210.752i	1.15165i
$184$	0	0
$185$	59.7512	0.322980
$186$	0	0
$187$	11.4930i	0.0614598i
$188$	0	0
$189$	−185.430	−0.981111
$190$	0	0
$191$	44.6569i	0.233806i	0.993143	+	0.116903i	$0.0372966\pi$
$191$	44.6569i	0.233806i	−0.993143	+	0.116903i	$0.962703\pi$
$192$	0	0
$193$	−85.7673	−0.444390	−0.222195	−	0.975002i	$-0.571322\pi$
$193$	−85.7673	−0.444390	−0.222195	+	0.975002i	$0.571322\pi$
$194$	0	0
$195$	− 38.3956i	− 0.196901i
$196$	0	0
$197$	−342.910	−1.74066	−0.870329	−	0.492471i	$-0.836094\pi$
$197$	−342.910	−1.74066	−0.870329	+	0.492471i	$0.836094\pi$
$198$	0	0
$199$	124.795i	0.627111i	0.949570	+	0.313555i	$0.101520\pi$
$199$	124.795i	0.627111i	−0.949570	+	0.313555i	$0.898480\pi$
$200$	0	0
$201$	−22.9737	−0.114297
$202$	0	0
$203$	9.39480i	0.0462798i
$204$	0	0
$205$	66.5989	0.324873
$206$	0	0
$207$	− 9.66174i	− 0.0466751i
$208$	0	0
$209$	22.1476	0.105969
$210$	0	0
$211$	81.1441i	0.384569i	0.981339	+	0.192285i	$0.0615897\pi$
$211$	81.1441i	0.384569i	−0.981339	+	0.192285i	$0.938410\pi$
$212$	0	0
$213$	−160.070	−0.751504
$214$	0	0
$215$	173.118i	0.805198i
$216$	0	0
$217$	−172.513	−0.794989
$218$	0	0
$219$	− 137.685i	− 0.628697i
$220$	0	0
$221$	−73.0824	−0.330690
$222$	0	0
$223$	261.174i	1.17118i	0.810607	+	0.585591i	$0.199138\pi$
$223$	261.174i	1.17118i	−0.810607	+	0.585591i	$0.800862\pi$
$224$	0	0
$225$	−10.0731	−0.0447691
$226$	0	0
$227$	− 88.4390i	− 0.389599i	−0.980843	−	0.194799i	$-0.937594\pi$
$227$	− 88.4390i	− 0.389599i	0.980843	−	0.194799i	$-0.0624056\pi$
$228$	0	0
$229$	−319.477	−1.39510	−0.697549	−	0.716537i	$-0.745726\pi$
$229$	−319.477	−1.39510	−0.697549	+	0.716537i	$0.745726\pi$
$230$	0	0
$231$	− 21.5984i	− 0.0934995i
$232$	0	0
$233$	76.2341	0.327185	0.163593	−	0.986528i	$-0.447692\pi$
$233$	76.2341	0.327185	0.163593	+	0.986528i	$0.447692\pi$
$234$	0	0
$235$	− 58.6245i	− 0.249466i
$236$	0	0
$237$	156.096	0.658633
$238$	0	0
$239$	163.387i	0.683629i	0.939768	+	0.341814i	$0.111041\pi$
$239$	163.387i	0.683629i	−0.939768	+	0.341814i	$0.888959\pi$
$240$	0	0
$241$	285.709	1.18552	0.592758	−	0.805381i	$-0.298039\pi$
$241$	285.709	1.18552	0.592758	+	0.805381i	$0.298039\pi$
$242$	0	0
$243$	106.881i	0.439838i
$244$	0	0
$245$	33.4848	0.136673
$246$	0	0
$247$	140.834i	0.570178i
$248$	0	0
$249$	502.850	2.01948
$250$	0	0
$251$	− 39.6896i	− 0.158126i	−0.996870	−	0.0790629i	$-0.974807\pi$
$251$	− 39.6896i	− 0.158126i	0.996870	−	0.0790629i	$-0.0251928\pi$
$252$	0	0
$253$	−3.90208	−0.0154232
$254$	0	0
$255$	104.826i	0.411084i
$256$	0	0
$257$	267.683	1.04157	0.520785	−	0.853688i	$-0.325639\pi$
$257$	267.683	1.04157	0.520785	+	0.853688i	$0.325639\pi$
$258$	0	0
$259$	213.731i	0.825215i
$260$	0	0
$261$	2.36632	0.00906637
$262$	0	0
$263$	− 83.6924i	− 0.318222i	−0.987261	−	0.159111i	$-0.949137\pi$
$263$	− 83.6924i	− 0.318222i	0.987261	−	0.159111i	$-0.0508628\pi$
$264$	0	0
$265$	−45.3324	−0.171066
$266$	0	0
$267$	182.515i	0.683578i
$268$	0	0
$269$	−56.6825	−0.210715	−0.105358	−	0.994434i	$-0.533599\pi$
$269$	−56.6825	−0.210715	−0.105358	+	0.994434i	$0.533599\pi$
$270$	0	0
$271$	− 54.0192i	− 0.199333i	−0.995021	−	0.0996664i	$-0.968222\pi$
$271$	− 54.0192i	− 0.199333i	0.995021	−	0.0996664i	$-0.0317776\pi$
$272$	0	0
$273$	137.341	0.503082
$274$	0	0
$275$	4.06820i	0.0147934i
$276$	0	0
$277$	29.1442	0.105214	0.0526068	−	0.998615i	$-0.483247\pi$
$277$	29.1442	0.105214	0.0526068	+	0.998615i	$0.483247\pi$
$278$	0	0
$279$	43.4518i	0.155741i
$280$	0	0
$281$	368.373	1.31094	0.655468	−	0.755223i	$-0.272471\pi$
$281$	368.373	1.31094	0.655468	+	0.755223i	$0.272471\pi$
$282$	0	0
$283$	− 305.703i	− 1.08022i	−0.841594	−	0.540111i	$-0.818382\pi$
$283$	− 305.703i	− 1.08022i	0.841594	−	0.540111i	$-0.181618\pi$
$284$	0	0
$285$	202.006	0.708793
$286$	0	0
$287$	238.225i	0.830052i
$288$	0	0
$289$	−89.4729	−0.309595
$290$	0	0
$291$	83.1450i	0.285722i
$292$	0	0
$293$	−543.617	−1.85535	−0.927675	−	0.373390i	$-0.878195\pi$
$293$	−543.617	−1.85535	−0.927675	+	0.373390i	$0.878195\pi$
$294$	0	0
$295$	− 41.6349i	− 0.141135i
$296$	0	0
$297$	18.8628	0.0635112
$298$	0	0
$299$	− 24.8128i	− 0.0829860i
$300$	0	0
$301$	−619.243	−2.05728
$302$	0	0
$303$	− 102.562i	− 0.338488i
$304$	0	0
$305$	141.995	0.465557
$306$	0	0
$307$	51.9157i	0.169106i	0.996419	+	0.0845532i	$0.0269463\pi$
$307$	51.9157i	0.169106i	−0.996419	+	0.0845532i	$0.973054\pi$
$308$	0	0
$309$	−115.783	−0.374701
$310$	0	0
$311$	196.302i	0.631195i	0.948893	+	0.315597i	$0.102205\pi$
$311$	196.302i	0.631195i	−0.948893	+	0.315597i	$0.897795\pi$
$312$	0	0
$313$	153.450	0.490256	0.245128	−	0.969491i	$-0.421170\pi$
$313$	153.450	0.490256	0.245128	+	0.969491i	$0.421170\pi$
$314$	0	0
$315$	− 36.0314i	− 0.114385i
$316$	0	0
$317$	360.488	1.13719	0.568594	−	0.822619i	$-0.307488\pi$
$317$	360.488	1.13719	0.568594	+	0.822619i	$0.307488\pi$
$318$	0	0
$319$	− 0.955685i	− 0.00299588i
$320$	0	0
$321$	353.622	1.10163
$322$	0	0
$323$	− 384.500i	− 1.19040i
$324$	0	0
$325$	−25.8691	−0.0795974
$326$	0	0
$327$	223.620i	0.683853i
$328$	0	0
$329$	209.700	0.637387
$330$	0	0
$331$	− 397.802i	− 1.20182i	−0.799317	−	0.600910i	$-0.794805\pi$
$331$	− 397.802i	− 1.20182i	0.799317	−	0.600910i	$-0.205195\pi$
$332$	0	0
$333$	53.8336	0.161662
$334$	0	0
$335$	15.4786i	0.0462047i
$336$	0	0
$337$	−383.698	−1.13857	−0.569285	−	0.822140i	$-0.692780\pi$
$337$	−383.698	−1.13857	−0.569285	+	0.822140i	$0.692780\pi$
$338$	0	0
$339$	− 93.2720i	− 0.275139i
$340$	0	0
$341$	17.5488	0.0514628
$342$	0	0
$343$	− 272.148i	− 0.793433i
$344$	0	0
$345$	−35.5905	−0.103161
$346$	0	0
$347$	57.8423i	0.166693i	0.996521	+	0.0833463i	$0.0265608\pi$
$347$	57.8423i	0.166693i	−0.996521	+	0.0833463i	$0.973439\pi$
$348$	0	0
$349$	−321.015	−0.919813	−0.459907	−	0.887967i	$-0.652117\pi$
$349$	−321.015	−0.919813	−0.459907	+	0.887967i	$0.652117\pi$
$350$	0	0
$351$	119.946i	0.341728i
$352$	0	0
$353$	−119.282	−0.337910	−0.168955	−	0.985624i	$-0.554039\pi$
$353$	−119.282	−0.337910	−0.168955	+	0.985624i	$0.554039\pi$
$354$	0	0
$355$	107.848i	0.303797i
$356$	0	0
$357$	−374.964	−1.05032
$358$	0	0
$359$	− 315.880i	− 0.879889i	−0.898025	−	0.439945i	$-0.854998\pi$
$359$	− 315.880i	− 0.879889i	0.898025	−	0.439945i	$-0.145002\pi$
$360$	0	0
$361$	−379.952	−1.05250
$362$	0	0
$363$	− 399.381i	− 1.10022i
$364$	0	0
$365$	−92.7654	−0.254152
$366$	0	0
$367$	430.949i	1.17425i	0.809497	+	0.587124i	$0.199740\pi$
$367$	430.949i	1.17425i	−0.809497	+	0.587124i	$0.800260\pi$
$368$	0	0
$369$	60.0031	0.162610
$370$	0	0
$371$	− 162.154i	− 0.437073i
$372$	0	0
$373$	−350.215	−0.938915	−0.469457	−	0.882955i	$-0.655550\pi$
$373$	−350.215	−0.938915	−0.469457	+	0.882955i	$0.655550\pi$
$374$	0	0
$375$	37.1056i	0.0989483i
$376$	0	0
$377$	6.07708	0.0161196
$378$	0	0
$379$	− 443.157i	− 1.16928i	−0.811293	−	0.584640i	$-0.801236\pi$
$379$	− 443.157i	− 1.16928i	0.811293	−	0.584640i	$-0.198764\pi$
$380$	0	0
$381$	730.560	1.91748
$382$	0	0
$383$	− 468.177i	− 1.22239i	−0.791479	−	0.611197i	$-0.790688\pi$
$383$	− 468.177i	− 1.22239i	0.791479	−	0.611197i	$-0.209312\pi$
$384$	0	0
$385$	−14.5520	−0.0377973
$386$	0	0
$387$	155.972i	0.403029i
$388$	0	0
$389$	−62.6706	−0.161107	−0.0805535	−	0.996750i	$-0.525669\pi$
$389$	−62.6706	−0.161107	−0.0805535	+	0.996750i	$0.525669\pi$
$390$	0	0
$391$	67.7431i	0.173256i
$392$	0	0
$393$	−366.242	−0.931912
$394$	0	0
$395$	− 105.170i	− 0.266253i
$396$	0	0
$397$	249.459	0.628361	0.314181	−	0.949363i	$-0.398270\pi$
$397$	249.459	0.628361	0.314181	+	0.949363i	$0.398270\pi$
$398$	0	0
$399$	722.577i	1.81097i
$400$	0	0
$401$	−139.397	−0.347624	−0.173812	−	0.984779i	$-0.555608\pi$
$401$	−139.397	−0.347624	−0.173812	+	0.984779i	$0.555608\pi$
$402$	0	0
$403$	111.591i	0.276900i
$404$	0	0
$405$	212.589	0.524912
$406$	0	0
$407$	− 21.7417i	− 0.0534194i
$408$	0	0
$409$	−332.198	−0.812220	−0.406110	−	0.913824i	$-0.633115\pi$
$409$	−332.198	−0.812220	−0.406110	+	0.913824i	$0.633115\pi$
$410$	0	0
$411$	504.940i	1.22856i
$412$	0	0
$413$	148.928	0.360601
$414$	0	0
$415$	− 338.796i	− 0.816376i
$416$	0	0
$417$	−236.725	−0.567685
$418$	0	0
$419$	− 327.954i	− 0.782706i	−0.920241	−	0.391353i	$-0.872007\pi$
$419$	− 327.954i	− 0.782706i	0.920241	−	0.391353i	$-0.127993\pi$
$420$	0	0
$421$	58.0927	0.137987	0.0689937	−	0.997617i	$-0.478021\pi$
$421$	58.0927	0.137987	0.0689937	+	0.997617i	$0.478021\pi$
$422$	0	0
$423$	− 52.8185i	− 0.124866i
$424$	0	0
$425$	70.6270	0.166181
$426$	0	0
$427$	507.917i	1.18950i
$428$	0	0
$429$	−13.9710	−0.0325665
$430$	0	0
$431$	− 272.197i	− 0.631547i	−0.948835	−	0.315774i	$-0.897736\pi$
$431$	− 272.197i	− 0.631547i	0.948835	−	0.315774i	$-0.102264\pi$
$432$	0	0
$433$	731.667	1.68976	0.844881	−	0.534954i	$-0.179671\pi$
$433$	731.667	1.68976	0.844881	+	0.534954i	$0.179671\pi$
$434$	0	0
$435$	− 8.71671i	− 0.0200384i
$436$	0	0
$437$	130.545	0.298729
$438$	0	0
$439$	259.180i	0.590386i	0.955438	+	0.295193i	$0.0953840\pi$
$439$	259.180i	0.590386i	−0.955438	+	0.295193i	$0.904616\pi$
$440$	0	0
$441$	30.1685	0.0684093
$442$	0	0
$443$	− 721.560i	− 1.62880i	−0.580301	−	0.814402i	$-0.697065\pi$
$443$	− 721.560i	− 1.62880i	0.580301	−	0.814402i	$-0.302935\pi$
$444$	0	0
$445$	122.970	0.276337
$446$	0	0
$447$	137.008i	0.306506i
$448$	0	0
$449$	94.1132	0.209606	0.104803	−	0.994493i	$-0.466579\pi$
$449$	94.1132	0.209606	0.104803	+	0.994493i	$0.466579\pi$
$450$	0	0
$451$	− 24.2334i	− 0.0537326i
$452$	0	0
$453$	−667.366	−1.47321
$454$	0	0
$455$	− 92.5341i	− 0.203372i
$456$	0	0
$457$	−75.2560	−0.164674	−0.0823370	−	0.996605i	$-0.526238\pi$
$457$	−75.2560	−0.164674	−0.0823370	+	0.996605i	$0.526238\pi$
$458$	0	0
$459$	− 327.473i	− 0.713450i
$460$	0	0
$461$	696.403	1.51064	0.755318	−	0.655358i	$-0.227482\pi$
$461$	696.403	1.51064	0.755318	+	0.655358i	$0.227482\pi$
$462$	0	0
$463$	− 262.748i	− 0.567491i	−0.958900	−	0.283745i	$-0.908423\pi$
$463$	− 262.748i	− 0.567491i	0.958900	−	0.283745i	$-0.0915771\pi$
$464$	0	0
$465$	160.061	0.344218
$466$	0	0
$467$	− 61.8446i	− 0.132430i	−0.997805	−	0.0662148i	$-0.978908\pi$
$467$	− 61.8446i	− 0.132430i	0.997805	−	0.0662148i	$-0.0210923\pi$
$468$	0	0
$469$	−55.3670	−0.118053
$470$	0	0
$471$	− 628.435i	− 1.33426i
$472$	0	0
$473$	62.9924	0.133176
$474$	0	0
$475$	− 136.102i	− 0.286531i
$476$	0	0
$477$	−40.8427	−0.0856242
$478$	0	0
$479$	− 792.764i	− 1.65504i	−0.561436	−	0.827520i	$-0.689751\pi$
$479$	− 792.764i	− 1.65504i	0.561436	−	0.827520i	$-0.310249\pi$
$480$	0	0
$481$	138.253	0.287428
$482$	0	0
$483$	− 127.307i	− 0.263576i
$484$	0	0
$485$	56.0192	0.115503
$486$	0	0
$487$	− 387.116i	− 0.794900i	−0.917624	−	0.397450i	$-0.869895\pi$
$487$	− 387.116i	− 0.794900i	0.917624	−	0.397450i	$-0.130105\pi$
$488$	0	0
$489$	242.850	0.496626
$490$	0	0
$491$	− 685.264i	− 1.39565i	−0.716269	−	0.697824i	$-0.754152\pi$
$491$	− 685.264i	− 1.39565i	0.716269	−	0.697824i	$-0.245848\pi$
$492$	0	0
$493$	−16.5914	−0.0336540
$494$	0	0
$495$	3.66529i	0.00740462i
$496$	0	0
$497$	−385.772	−0.776202
$498$	0	0
$499$	− 506.495i	− 1.01502i	−0.861646	−	0.507510i	$-0.830566\pi$
$499$	− 506.495i	− 1.01502i	0.861646	−	0.507510i	$-0.169434\pi$
$500$	0	0
$501$	891.354	1.77915
$502$	0	0
$503$	155.512i	0.309168i	0.987980	+	0.154584i	$0.0494038\pi$
$503$	155.512i	0.309168i	−0.987980	+	0.154584i	$0.950596\pi$
$504$	0	0
$505$	−69.1013	−0.136834
$506$	0	0
$507$	472.042i	0.931049i
$508$	0	0
$509$	−585.180	−1.14967	−0.574833	−	0.818271i	$-0.694933\pi$
$509$	−585.180	−1.14967	−0.574833	+	0.818271i	$0.694933\pi$
$510$	0	0
$511$	− 331.823i	− 0.649359i
$512$	0	0
$513$	−631.059	−1.23014
$514$	0	0
$515$	78.0089i	0.151474i
$516$	0	0
$517$	−21.3317	−0.0412606
$518$	0	0
$519$	− 411.152i	− 0.792200i
$520$	0	0
$521$	299.033	0.573960	0.286980	−	0.957937i	$-0.407349\pi$
$521$	299.033	0.573960	0.286980	+	0.957937i	$0.407349\pi$
$522$	0	0
$523$	204.620i	0.391242i	0.980680	+	0.195621i	$0.0626723\pi$
$523$	204.620i	0.391242i	−0.980680	+	0.195621i	$0.937328\pi$
$524$	0	0
$525$	−132.727	−0.252813
$526$	0	0
$527$	− 304.661i	− 0.578105i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	− 37.5115i	− 0.0706430i
$532$	0	0
$533$	154.097	0.289113
$534$	0	0
$535$	− 238.254i	− 0.445334i
$536$	0	0
$537$	−510.159	−0.950017
$538$	0	0
$539$	− 12.1841i	− 0.0226051i
$540$	0	0
$541$	905.287	1.67336	0.836679	−	0.547693i	$-0.184494\pi$
$541$	905.287	1.67336	0.836679	+	0.547693i	$0.184494\pi$
$542$	0	0
$543$	516.216i	0.950675i
$544$	0	0
$545$	150.664	0.276449
$546$	0	0
$547$	− 833.512i	− 1.52379i	−0.647702	−	0.761894i	$-0.724270\pi$
$547$	− 833.512i	− 1.52379i	0.647702	−	0.761894i	$-0.275730\pi$
$548$	0	0
$549$	127.932	0.233027
$550$	0	0
$551$	31.9726i	0.0580265i
$552$	0	0
$553$	376.194	0.680278
$554$	0	0
$555$	− 198.304i	− 0.357305i
$556$	0	0
$557$	−664.115	−1.19231	−0.596153	−	0.802871i	$-0.703305\pi$
$557$	−664.115	−1.19231	−0.596153	+	0.802871i	$0.703305\pi$
$558$	0	0
$559$	400.561i	0.716567i
$560$	0	0
$561$	38.1432	0.0679915
$562$	0	0
$563$	104.785i	0.186119i	0.995661	+	0.0930594i	$0.0296647\pi$
$563$	104.785i	0.186119i	−0.995661	+	0.0930594i	$0.970335\pi$
$564$	0	0
$565$	−62.8423	−0.111225
$566$	0	0
$567$	760.433i	1.34115i
$568$	0	0
$569$	605.766	1.06462	0.532308	−	0.846551i	$-0.321325\pi$
$569$	605.766	1.06462	0.532308	+	0.846551i	$0.321325\pi$
$570$	0	0
$571$	− 981.680i	− 1.71923i	−0.510942	−	0.859615i	$-0.670703\pi$
$571$	− 981.680i	− 1.71923i	0.510942	−	0.859615i	$-0.329297\pi$
$572$	0	0
$573$	148.209	0.258654
$574$	0	0
$575$	23.9792i	0.0417029i
$576$	0	0
$577$	−284.541	−0.493139	−0.246569	−	0.969125i	$-0.579303\pi$
$577$	−284.541	−0.493139	−0.246569	+	0.969125i	$0.579303\pi$
$578$	0	0
$579$	284.647i	0.491618i
$580$	0	0
$581$	1211.88	2.08584
$582$	0	0
$583$	16.4951i	0.0282935i
$584$	0	0
$585$	−23.3071	−0.0398412
$586$	0	0
$587$	23.9032i	0.0407210i	0.999793	+	0.0203605i	$0.00648139\pi$
$587$	23.9032i	0.0407210i	−0.999793	+	0.0203605i	$0.993519\pi$
$588$	0	0
$589$	−587.099	−0.996772
$590$	0	0
$591$	1138.06i	1.92565i
$592$	0	0
$593$	997.581	1.68226	0.841130	−	0.540832i	$-0.181891\pi$
$593$	997.581	1.68226	0.841130	+	0.540832i	$0.181891\pi$
$594$	0	0
$595$	252.633i	0.424594i
$596$	0	0
$597$	414.173	0.693758
$598$	0	0
$599$	656.332i	1.09571i	0.836572	+	0.547857i	$0.184556\pi$
$599$	656.332i	1.09571i	−0.836572	+	0.547857i	$0.815444\pi$
$600$	0	0
$601$	−49.6459	−0.0826054	−0.0413027	−	0.999147i	$-0.513151\pi$
$601$	−49.6459	−0.0826054	−0.0413027	+	0.999147i	$0.513151\pi$
$602$	0	0
$603$	13.9456i	0.0231271i
$604$	0	0
$605$	−269.084	−0.444767
$606$	0	0
$607$	− 376.522i	− 0.620300i	−0.950688	−	0.310150i	$-0.899621\pi$
$607$	− 376.522i	− 0.620300i	0.950688	−	0.310150i	$-0.100379\pi$
$608$	0	0
$609$	31.1797	0.0511982
$610$	0	0
$611$	− 135.646i	− 0.222006i
$612$	0	0
$613$	−641.260	−1.04610	−0.523051	−	0.852302i	$-0.675206\pi$
$613$	−641.260	−1.04610	−0.523051	+	0.852302i	$0.675206\pi$
$614$	0	0
$615$	− 221.030i	− 0.359399i
$616$	0	0
$617$	−291.345	−0.472195	−0.236098	−	0.971729i	$-0.575869\pi$
$617$	−291.345	−0.472195	−0.236098	+	0.971729i	$0.575869\pi$
$618$	0	0
$619$	1157.46i	1.86988i	0.354802	+	0.934941i	$0.384548\pi$
$619$	1157.46i	1.86988i	−0.354802	+	0.934941i	$0.615452\pi$
$620$	0	0
$621$	111.183	0.179039
$622$	0	0
$623$	439.865i	0.706043i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	− 73.5041i	− 0.117231i
$628$	0	0
$629$	−377.453	−0.600084
$630$	0	0
$631$	− 476.562i	− 0.755249i	−0.925959	−	0.377624i	$-0.876741\pi$
$631$	− 476.562i	− 0.755249i	0.925959	−	0.377624i	$-0.123259\pi$
$632$	0	0
$633$	269.303	0.425440
$634$	0	0
$635$	− 492.217i	− 0.775144i
$636$	0	0
$637$	77.4773	0.121628
$638$	0	0
$639$	97.1667i	0.152061i
$640$	0	0
$641$	367.203	0.572860	0.286430	−	0.958101i	$-0.407531\pi$
$641$	367.203	0.572860	0.286430	+	0.958101i	$0.407531\pi$
$642$	0	0
$643$	47.6691i	0.0741355i	0.999313	+	0.0370677i	$0.0118017\pi$
$643$	47.6691i	0.0741355i	−0.999313	+	0.0370677i	$0.988198\pi$
$644$	0	0
$645$	574.547	0.890771
$646$	0	0
$647$	324.514i	0.501568i	0.968043	+	0.250784i	$0.0806884\pi$
$647$	324.514i	0.501568i	−0.968043	+	0.250784i	$0.919312\pi$
$648$	0	0
$649$	−15.1497	−0.0233432
$650$	0	0
$651$	572.540i	0.879477i
$652$	0	0
$653$	587.705	0.900008	0.450004	−	0.893027i	$-0.351423\pi$
$653$	587.705	0.900008	0.450004	+	0.893027i	$0.351423\pi$
$654$	0	0
$655$	246.756i	0.376727i
$656$	0	0
$657$	−83.5781	−0.127212
$658$	0	0
$659$	− 816.724i	− 1.23934i	−0.784863	−	0.619669i	$-0.787267\pi$
$659$	− 816.724i	− 1.23934i	0.784863	−	0.619669i	$-0.212733\pi$
$660$	0	0
$661$	1181.50	1.78745	0.893724	−	0.448618i	$-0.148084\pi$
$661$	1181.50	1.78745	0.893724	+	0.448618i	$0.148084\pi$
$662$	0	0
$663$	242.548i	0.365834i
$664$	0	0
$665$	486.838	0.732088
$666$	0	0
$667$	− 5.63309i	− 0.00844541i
$668$	0	0
$669$	866.790	1.29565
$670$	0	0
$671$	− 51.6677i	− 0.0770011i
$672$	0	0
$673$	673.043	1.00006	0.500032	−	0.866007i	$-0.333322\pi$
$673$	673.043	1.00006	0.500032	+	0.866007i	$0.333322\pi$
$674$	0	0
$675$	− 115.916i	− 0.171728i
$676$	0	0
$677$	1159.88	1.71326	0.856629	−	0.515934i	$-0.172555\pi$
$677$	1159.88	1.71326	0.856629	+	0.515934i	$0.172555\pi$
$678$	0	0
$679$	200.381i	0.295112i
$680$	0	0
$681$	−293.514	−0.431004
$682$	0	0
$683$	710.065i	1.03963i	0.854280	+	0.519813i	$0.173998\pi$
$683$	710.065i	1.03963i	−0.854280	+	0.519813i	$0.826002\pi$
$684$	0	0
$685$	340.205	0.496649
$686$	0	0
$687$	1060.29i	1.54336i
$688$	0	0
$689$	−104.890	−0.152236
$690$	0	0
$691$	76.8999i	0.111288i	0.998451	+	0.0556440i	$0.0177212\pi$
$691$	76.8999i	0.111288i	−0.998451	+	0.0556440i	$0.982279\pi$
$692$	0	0
$693$	−13.1108	−0.0189188
$694$	0	0
$695$	159.494i	0.229488i
$696$	0	0
$697$	−420.710	−0.603602
$698$	0	0
$699$	− 253.008i	− 0.361957i
$700$	0	0
$701$	433.594	0.618537	0.309268	−	0.950975i	$-0.399916\pi$
$701$	433.594	0.618537	0.309268	+	0.950975i	$0.399916\pi$
$702$	0	0
$703$	727.373i	1.03467i
$704$	0	0
$705$	−194.565	−0.275978
$706$	0	0
$707$	− 247.176i	− 0.349612i
$708$	0	0
$709$	857.201	1.20903	0.604514	−	0.796594i	$-0.293367\pi$
$709$	857.201	1.20903	0.604514	+	0.796594i	$0.293367\pi$
$710$	0	0
$711$	− 94.7542i	− 0.133269i
$712$	0	0
$713$	103.438	0.145074
$714$	0	0
$715$	9.41302i	0.0131651i
$716$	0	0
$717$	542.254	0.756282
$718$	0	0
$719$	− 291.699i	− 0.405702i	−0.979210	−	0.202851i	$-0.934979\pi$
$719$	− 291.699i	− 0.405702i	0.979210	−	0.202851i	$-0.0650206\pi$
$720$	0	0
$721$	−279.038	−0.387016
$722$	0	0
$723$	− 948.219i	− 1.31151i
$724$	0	0
$725$	−5.87290	−0.00810056
$726$	0	0
$727$	256.942i	0.353428i	0.984262	+	0.176714i	$0.0565468\pi$
$727$	256.942i	0.353428i	−0.984262	+	0.176714i	$0.943453\pi$
$728$	0	0
$729$	−500.937	−0.687157
$730$	0	0
$731$	− 1093.60i	− 1.49603i
$732$	0	0
$733$	971.976	1.32602	0.663012	−	0.748609i	$-0.269278\pi$
$733$	971.976	1.32602	0.663012	+	0.748609i	$0.269278\pi$
$734$	0	0
$735$	− 111.130i	− 0.151198i
$736$	0	0
$737$	5.63220	0.00764206
$738$	0	0
$739$	559.385i	0.756948i	0.925612	+	0.378474i	$0.123551\pi$
$739$	559.385i	0.756948i	−0.925612	+	0.378474i	$0.876449\pi$
$740$	0	0
$741$	467.403	0.630774
$742$	0	0
$743$	− 519.999i	− 0.699864i	−0.936775	−	0.349932i	$-0.886205\pi$
$743$	− 519.999i	− 0.699864i	0.936775	−	0.349932i	$-0.113795\pi$
$744$	0	0
$745$	92.3097	0.123906
$746$	0	0
$747$	− 305.242i	− 0.408624i
$748$	0	0
$749$	852.235	1.13783
$750$	0	0
$751$	191.072i	0.254424i	0.991876	+	0.127212i	$0.0406029\pi$
$751$	191.072i	0.254424i	−0.991876	+	0.127212i	$0.959397\pi$
$752$	0	0
$753$	−131.723	−0.174931
$754$	0	0
$755$	449.639i	0.595549i
$756$	0	0
$757$	383.809	0.507013	0.253507	−	0.967334i	$-0.418416\pi$
$757$	383.809	0.507013	0.253507	+	0.967334i	$0.418416\pi$
$758$	0	0
$759$	12.9503i	0.0170623i
$760$	0	0
$761$	773.453	1.01636	0.508182	−	0.861250i	$-0.330318\pi$
$761$	773.453	1.01636	0.508182	+	0.861250i	$0.330318\pi$
$762$	0	0
$763$	538.928i	0.706327i
$764$	0	0
$765$	63.6322	0.0831794
$766$	0	0
$767$	− 96.3351i	− 0.125600i
$768$	0	0
$769$	51.7401	0.0672823	0.0336411	−	0.999434i	$-0.489290\pi$
$769$	51.7401	0.0672823	0.0336411	+	0.999434i	$0.489290\pi$
$770$	0	0
$771$	− 888.395i	− 1.15226i
$772$	0	0
$773$	−473.049	−0.611965	−0.305982	−	0.952037i	$-0.598985\pi$
$773$	−473.049	−0.611965	−0.305982	+	0.952037i	$0.598985\pi$
$774$	0	0
$775$	− 107.842i	− 0.139150i
$776$	0	0
$777$	709.335	0.912915
$778$	0	0
$779$	810.732i	1.04073i
$780$	0	0
$781$	39.2426	0.0502466
$782$	0	0
$783$	27.2307i	0.0347773i
$784$	0	0
$785$	−423.410	−0.539376
$786$	0	0
$787$	− 503.773i	− 0.640118i	−0.947398	−	0.320059i	$-0.896297\pi$
$787$	− 503.773i	− 0.640118i	0.947398	−	0.320059i	$-0.103703\pi$
$788$	0	0
$789$	−277.760	−0.352041
$790$	0	0
$791$	− 224.787i	− 0.284181i
$792$	0	0
$793$	328.549	0.414311
$794$	0	0
$795$	150.450i	0.189246i
$796$	0	0
$797$	−1065.65	−1.33708	−0.668539	−	0.743677i	$-0.733080\pi$
$797$	−1065.65	−1.33708	−0.668539	+	0.743677i	$0.733080\pi$
$798$	0	0
$799$	370.336i	0.463499i
$800$	0	0
$801$	110.791	0.138316
$802$	0	0
$803$	33.7546i	0.0420356i
$804$	0	0
$805$	−85.7736	−0.106551
$806$	0	0
$807$	188.119i	0.233109i
$808$	0	0
$809$	−652.330	−0.806341	−0.403170	−	0.915125i	$-0.632092\pi$
$809$	−652.330	−0.806341	−0.403170	+	0.915125i	$0.632092\pi$
$810$	0	0
$811$	− 569.949i	− 0.702773i	−0.936230	−	0.351387i	$-0.885710\pi$
$811$	− 569.949i	− 0.702773i	0.936230	−	0.351387i	$-0.114290\pi$
$812$	0	0
$813$	−179.280	−0.220517
$814$	0	0
$815$	− 163.621i	− 0.200762i
$816$	0	0
$817$	−2107.42	−2.57946
$818$	0	0
$819$	− 83.3697i	− 0.101794i
$820$	0	0
$821$	−1384.79	−1.68671	−0.843356	−	0.537355i	$-0.819423\pi$
$821$	−1384.79	−1.68671	−0.843356	+	0.537355i	$0.819423\pi$
$822$	0	0
$823$	462.653i	0.562155i	0.959685	+	0.281077i	$0.0906918\pi$
$823$	462.653i	0.562155i	−0.959685	+	0.281077i	$0.909308\pi$
$824$	0	0
$825$	13.5016	0.0163656
$826$	0	0
$827$	− 34.8661i	− 0.0421597i	−0.999778	−	0.0210799i	$-0.993290\pi$
$827$	− 34.8661i	− 0.0421597i	0.999778	−	0.0210799i	$-0.00671042\pi$
$828$	0	0
$829$	987.912	1.19169	0.595846	−	0.803099i	$-0.296817\pi$
$829$	987.912	1.19169	0.595846	+	0.803099i	$0.296817\pi$
$830$	0	0
$831$	− 96.7245i	− 0.116395i
$832$	0	0
$833$	−211.526	−0.253933
$834$	0	0
$835$	− 600.552i	− 0.719224i
$836$	0	0
$837$	−500.025	−0.597401
$838$	0	0
$839$	159.837i	0.190509i	0.995453	+	0.0952547i	$0.0303665\pi$
$839$	159.837i	0.190509i	−0.995453	+	0.0952547i	$0.969633\pi$
$840$	0	0
$841$	−839.620	−0.998360
$842$	0	0
$843$	− 1222.57i	− 1.45026i
$844$	0	0
$845$	318.039	0.376378
$846$	0	0
$847$	− 962.515i	− 1.13638i
$848$	0	0
$849$	−1014.57	−1.19502
$850$	0	0
$851$	− 128.152i	− 0.150590i
$852$	0	0
$853$	−1322.98	−1.55097	−0.775485	−	0.631366i	$-0.782495\pi$
$853$	−1322.98	−1.55097	−0.775485	+	0.631366i	$0.782495\pi$
$854$	0	0
$855$	− 122.623i	− 0.143418i
$856$	0	0
$857$	772.698	0.901631	0.450815	−	0.892617i	$-0.351133\pi$
$857$	772.698	0.901631	0.450815	+	0.892617i	$0.351133\pi$
$858$	0	0
$859$	264.420i	0.307823i	0.988085	+	0.153912i	$0.0491871\pi$
$859$	264.420i	0.307823i	−0.988085	+	0.153912i	$0.950813\pi$
$860$	0	0
$861$	790.627	0.918266
$862$	0	0
$863$	− 27.2201i	− 0.0315412i	−0.999876	−	0.0157706i	$-0.994980\pi$
$863$	− 27.2201i	− 0.0315412i	0.999876	−	0.0157706i	$-0.00502015\pi$
$864$	0	0
$865$	−277.015	−0.320248
$866$	0	0
$867$	296.945i	0.342497i
$868$	0	0
$869$	−38.2683	−0.0440372
$870$	0	0
$871$	35.8145i	0.0411188i
$872$	0	0
$873$	50.4711	0.0578134
$874$	0	0
$875$	89.4251i	0.102200i
$876$	0	0
$877$	−477.754	−0.544759	−0.272380	−	0.962190i	$-0.587811\pi$
$877$	−477.754	−0.544759	−0.272380	+	0.962190i	$0.587811\pi$
$878$	0	0
$879$	1804.17i	2.05253i
$880$	0	0
$881$	−297.317	−0.337477	−0.168739	−	0.985661i	$-0.553969\pi$
$881$	−297.317	−0.337477	−0.168739	+	0.985661i	$0.553969\pi$
$882$	0	0
$883$	1508.08i	1.70790i	0.520352	+	0.853952i	$0.325801\pi$
$883$	1508.08i	1.70790i	−0.520352	+	0.853952i	$0.674199\pi$
$884$	0	0
$885$	−138.179	−0.156135
$886$	0	0
$887$	− 1342.19i	− 1.51318i	−0.653887	−	0.756592i	$-0.726863\pi$
$887$	− 1342.19i	− 1.51318i	0.653887	−	0.756592i	$-0.273137\pi$
$888$	0	0
$889$	1760.66	1.98050
$890$	0	0
$891$	− 77.3550i	− 0.0868182i
$892$	0	0
$893$	713.657	0.799168
$894$	0	0
$895$	343.721i	0.384046i
$896$	0	0
$897$	−82.3494	−0.0918054
$898$	0	0
$899$	25.3337i	0.0281799i
$900$	0	0
$901$	286.368	0.317833
$902$	0	0
$903$	2055.16i	2.27592i
$904$	0	0
$905$	347.802	0.384312
$906$	0	0
$907$	− 1104.27i	− 1.21750i	−0.793362	−	0.608750i	$-0.791671\pi$
$907$	− 1104.27i	− 1.21750i	0.793362	−	0.608750i	$-0.208329\pi$
$908$	0	0
$909$	−62.2576	−0.0684902
$910$	0	0
$911$	− 133.328i	− 0.146353i	−0.997319	−	0.0731767i	$-0.976686\pi$
$911$	− 133.328i	− 0.146353i	0.997319	−	0.0731767i	$-0.0233137\pi$
$912$	0	0
$913$	−123.278	−0.135025
$914$	0	0
$915$	− 471.256i	− 0.515034i
$916$	0	0
$917$	−882.648	−0.962539
$918$	0	0
$919$	1337.48i	1.45536i	0.685916	+	0.727681i	$0.259402\pi$
$919$	1337.48i	1.45536i	−0.685916	+	0.727681i	$0.740598\pi$
$920$	0	0
$921$	172.299	0.187078
$922$	0	0
$923$	249.539i	0.270356i
$924$	0	0
$925$	−133.608	−0.144441
$926$	0	0
$927$	70.2830i	0.0758177i
$928$	0	0
$929$	−1123.15	−1.20899	−0.604496	−	0.796608i	$-0.706626\pi$
$929$	−1123.15	−1.20899	−0.604496	+	0.796608i	$0.706626\pi$
$930$	0	0
$931$	407.622i	0.437832i
$932$	0	0
$933$	651.491	0.698276
$934$	0	0
$935$	− 25.6991i	− 0.0274857i
$936$	0	0
$937$	−1214.23	−1.29587	−0.647937	−	0.761694i	$-0.724368\pi$
$937$	−1214.23	−1.29587	−0.647937	+	0.761694i	$0.724368\pi$
$938$	0	0
$939$	− 509.275i	− 0.542359i
$940$	0	0
$941$	617.170	0.655866	0.327933	−	0.944701i	$-0.393648\pi$
$941$	617.170	0.655866	0.327933	+	0.944701i	$0.393648\pi$
$942$	0	0
$943$	− 142.839i	− 0.151473i
$944$	0	0
$945$	414.634	0.438766
$946$	0	0
$947$	247.277i	0.261116i	0.991441	+	0.130558i	$0.0416768\pi$
$947$	247.277i	0.261116i	−0.991441	+	0.130558i	$0.958323\pi$
$948$	0	0
$949$	−214.641	−0.226176
$950$	0	0
$951$	− 1196.40i	− 1.25804i
$952$	0	0
$953$	470.962	0.494189	0.247094	−	0.968991i	$-0.420524\pi$
$953$	470.962	0.494189	0.247094	+	0.968991i	$0.420524\pi$
$954$	0	0
$955$	− 99.8559i	− 0.104561i
$956$	0	0
$957$	−3.17175	−0.00331427
$958$	0	0
$959$	1216.91i	1.26894i
$960$	0	0
$961$	495.808	0.515929
$962$	0	0
$963$	− 214.657i	− 0.222905i
$964$	0	0
$965$	191.781	0.198737
$966$	0	0
$967$	− 427.028i	− 0.441601i	−0.975319	−	0.220801i	$-0.929133\pi$
$967$	− 427.028i	− 0.441601i	0.975319	−	0.220801i	$-0.0708670\pi$
$968$	0	0
$969$	−1276.09	−1.31691
$970$	0	0
$971$	− 530.733i	− 0.546583i	−0.961931	−	0.273292i	$-0.911888\pi$
$971$	− 530.733i	− 0.546583i	0.961931	−	0.273292i	$-0.0881124\pi$
$972$	0	0
$973$	−570.511	−0.586342
$974$	0	0
$975$	85.8552i	0.0880566i
$976$	0	0
$977$	−1325.68	−1.35689	−0.678447	−	0.734650i	$-0.737347\pi$
$977$	−1325.68	−1.35689	−0.678447	+	0.734650i	$0.737347\pi$
$978$	0	0
$979$	− 44.7452i	− 0.0457050i
$980$	0	0
$981$	135.743	0.138372
$982$	0	0
$983$	− 1413.19i	− 1.43763i	−0.695200	−	0.718817i	$-0.744684\pi$
$983$	− 1413.19i	− 1.43763i	0.695200	−	0.718817i	$-0.255316\pi$
$984$	0	0
$985$	766.769	0.778446
$986$	0	0
$987$	− 695.959i	− 0.705126i
$988$	0	0
$989$	371.296	0.375426
$990$	0	0
$991$	351.073i	0.354261i	0.984187	+	0.177131i	$0.0566815\pi$
$991$	351.073i	0.354261i	−0.984187	+	0.177131i	$0.943318\pi$
$992$	0	0
$993$	−1320.24	−1.32954
$994$	0	0
$995$	− 279.050i	− 0.280453i
$996$	0	0
$997$	−757.850	−0.760130	−0.380065	−	0.924960i	$-0.624098\pi$
$997$	−757.850	−0.760130	−0.380065	+	0.924960i	$0.624098\pi$
$998$	0	0
$999$	619.494i	0.620114i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.14	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.43	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.14	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.43	yes	56	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.31883i q^{3} -2.23607 q^{5} -7.99843i q^{7} -2.01461 q^{9} +O(q^{10})\)	\(q-3.31883i q^{3} -2.23607 q^{5} -7.99843i q^{7} -2.01461 q^{9} +0.813639i q^{11} -5.17383 q^{13} +7.42112i q^{15} +14.1254 q^{17} -27.2204i q^{19} -26.5454 q^{21} +4.79583i q^{23} +5.00000 q^{25} -23.1833i q^{27} -1.17458 q^{29} -21.5683i q^{31} +2.70033 q^{33} +17.8850i q^{35} -26.7216 q^{37} +17.1710i q^{39} -29.7840 q^{41} -77.4205i q^{43} +4.50481 q^{45} +26.2177i q^{47} -14.9749 q^{49} -46.8798i q^{51} +20.2733 q^{53} -1.81935i q^{55} -90.3399 q^{57} +18.6197i q^{59} -63.5020 q^{61} +16.1137i q^{63} +11.5690 q^{65} -6.92224i q^{67} +15.9165 q^{69} -48.2310i q^{71} +41.4860 q^{73} -16.5941i q^{75} +6.50783 q^{77} +47.0335i q^{79} -95.0728 q^{81} +151.514i q^{83} -31.5854 q^{85} +3.89823i q^{87} -54.9939 q^{89} +41.3825i q^{91} -71.5815 q^{93} +60.8667i q^{95} -25.0525 q^{97} -1.63917i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 56 * q - 120 * q^9	\( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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